How are animations done in blender?
You would just need to add another keyframe for the second object that should stay on its current position until the first moving object has finished its movement. Otherwise the position of your second object would also be interpolated between the keyframes.

In this example the location of both objects is keyframed at frame 1 the sphere reaches its final location at frame 20. The cube stops at frame 42. To have the cube stand still another keyframe for the cubes starting position is added at frame 20.